Lowness properties and approximations of the jump

We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informally, a set A is strongly jump-traceable if for each order function h , for each input e one may effectively enumerate a set T e of possible values for the jump J A ( e ) , and the number of values enumerated is at most h ( e ) . A ′ is well-approximable if can be effectively approximated with less than h ( x ) changes at input x , for each order function h . We prove that there is a strongly jump-traceable set which is not computable, and that if A ′ is well-approximable then A is strongly jump-traceable. For r.e. sets, the converse holds as well. We characterize jump-traceability and strong jump-traceability in terms of Kolmogorov complexity. We also investigate other properties of these lowness properties.